Experimental and theoretical investigations of quantum state transfer and decoherence processes in quasi-one-dimensional systems in multiple-quantum NMR experiments

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by  G. A. Bochkin

Experimental and theoretical investigations of quantum state transfer and decoherence
processes in quasi-one-dimensional systems in multiple-quantum NMR experiments
G. A. Bochkin1, S. I. Doronin1, S. G. Vasil’ev1, A. V. Fedorova1, E. B. Fel’dman1
1 Institute of Problems of Chemical Physics, Chernogolovka, Moscow Region, 142432, Russia
Multiplequantum(MQ)NMRmethods[1]areappliedtotheanalysisofvariousproblemsofquan-
tuminformationprocessing. Itisshownthatthetwo-spin/two-quantumHamiltonian[1]describing
MQ NMR dynamics is related to the flip-flop Hamiltonian of a one-dimensional spin system in the
approximation of the nearest neighbor interactions. As a result, it is possible to organize quantum
state transfer along a linear chain. MQ NMR experiments are performed on quasi-one-dimensional
chainsof19FnucleiincalciumfluorapatiteCa (PO ) F.RelaxationoftheMQNMRcoherencesis
5 4 3
consideredasthesimplestmodelofdecoherenceprocesses. Atheoryofthedipolarrelaxationofthe
MQNMRcoherencesinone-dimensionalsystemsisdeveloped. Agoodagreementofthetheoretical
7 predictions and the experimental data is obtained.
1
0
2
I. INTRODUCTION

n
a
Thedevelopmentofexperimentalandtheoreticalmethodsofquantuminformationprocessing(QIP)isanimportant
J
direction of quantum informatics. NMR methods are the simplest ones among different experimental approaches [2].
3
ItiswellknownthattheliquidstateNMRmethods[3]arewidelyusedforexperimentalrealizationsofquantumgates
1
and algorithms on the basis of the pseudo-pure states [4, 5]. However, the possibilities of QIP based on the liquid

] state NMR are limitedbecause the number of correlated qubitsis verysmall (about 10) [6]and quantum correlations
h in such systems are not strong. In particular, entanglement is almost absent [7].
p
At the same time, the potential of the solid state NMR methods [8] is not exhausted yet. Multiple quantum (MQ)
-
t spectroscopy in solids [1] is an example of such a method. MQ NMR not only creates multi-qubit coherent states
n
but also allows the investigation of their relaxation under the action of the correlated spin reservoir. MQ NMR [1]
a
is an important method for the investigation of various problems of quantum information processing such as the
u
q transmission of quantum information [9] and decoherence processes [10, 11].
[ One-dimensional MQ NMR methods are very suitable for solving problems of quantum informatics. The reason

is that a consistent quantum-mechanical theory of MQ NMR dynamics has been developed only for one-dimensional
1
systems [12–15]. That theory is based on the fact that the non-secular two-spin/two-quantum Hamiltonian [1],
v
8 describing MQ NMR dynamics, is the XY Hamiltonian [16], which can be diagonalized exactly for one-dimensional
8 systems in the approximation of the nearest neighbor interactions [17]. As a result, MQ NMR dynamics in such
5 systems can be studied analytically. In particular, only MQ NMR coherences of the zeroth and plus/minus second
3 orders arise in a one-dimensional chain initially prepared in a thermodynamic equilibrium state [12–15].
0
Thedevelopedtheory[12–15]isbasedonthemodelofanisolatedspinchainwiththenearestneighborinteractions.
.
1 Such a model is realized in quasi-one-dimensional chains of 19F nuclei in calcium fluorapatite Ca (PO ) F with a
5 4 3
0 hexagonal system of fluorine nuclei. The distance between neighboring chains is about three times larger than the
7
distance between the nearest fluorine nuclei in the same chain. Taking into account that the dipole-dipole interaction
1
(DDI)ofthenextnearestspinsinthechainiseighttimesweakerthanthenearestneighborinteraction[17]onecanuse
:
v the model of an isolated spin chain with the nearest neighbor interactions for the interpretation of the experimental
i
X data obtained from the MQ NMR experiments [1].
The information resource of MQ NMR exceeds the resource of conventional NMR [15]. However, that resource can
r
a be used only if the XY Hamiltonian is created with sufficient accuracy [18]. The XY Hamiltonian is obtained as the
averagedtwo-spin/two-quantumHamiltonian[1]whichistheaverageHamiltonian[19]intheMQNMRexperiments.
The corrections to the average Hamiltonian [19] spoil the XY Hamiltonian. These corrections are proportional to
powers of the parameter (cid:15)=t ω (t is the period of the irradiating sequence of the preparation period of the MQ
c loc c
NMR experiment and ω is of the order of the DDI). We have designed a special probe [18] for the generation of
loc
high-power ultra-short pulses and decreased the period t and the parameter (cid:15) in MQ NMR experiments. The results
c
[18] show high accuracy of creating the XY Hamiltonian.
ItisveryimportantthattheXYHamiltonian(theMQNMRHamiltonian)isrelatedtotheflip-floponebyasimple
unitary transformation [14] for one-dimensional systems in the approximation of the nearest neighbor interactions.
That fact allows us to realize the quantum state transfer along a one-dimensional spin chain. It is also significant
that the relaxation of the MQ NMR coherences can be considered as a model for studying the decoherence processes.
The paper is organized as follows. An introduction to MQ NMR dynamics is given in section 2. In section 3 we
show that MQ NMR can be used for the quantum state transfer in one-dimensional spin chains. Relaxation of the
FIG. 1: Scheme of the multiple quantum NMR experiment. The nonsecular two-spin/two-quantum Hamiltonians H(0), H(0)
Φ
aredeterminedbyEqs. (1,4);τ isthedurationofthepreparationandmixingperiods;t andt aredurationsoftheevolution
1 2
anddetectingperiods,respectively. Thebasiccycleofthemulti-pulsesequenceconsistingofeightπ/2pulseswiththeduration
t . The pulses are separated by delays ∆ and ∆(cid:48) = 2∆+t , repeated m times for obtaining the necessary time τ = 12m
p p
(∆+t ); m is a positive integer.
p
MQ NMR coherences in one-dimensional spin chains is discussed in section 4. We briefly summarize our results in
section 5.
II. MQ NMR DYNAMICS OF ONE-DIMENSIONAL SPIN SYSTEMS
The MQ NMR experiment [1] consists of four distinct periods of time (Fig. 1): preparation(τ), evolution (t),
mixing(τ) and detection. The spin system is irradiated by a periodic sequence of resonance radio-frequency pulses
on the preparation period. As a result, the anisotropic DDI [20] become rapidly oscillating if the inverse period of
the irradiating sequence t−1 strongly exceeds the magnitude of the interactions ω ((cid:15)=t ω (cid:28)1). The theory of
c loc c loc
the average Hamiltonian [19] makes it possible to find the averaged interactions, the Hamiltonian of which H(0) (the
averaged non-secular two-spin/two-quantum Hamiltonian [1]) up to the terms of the order of (cid:15)2 has the form
H(0) =H(+2)+H(−2), (1)
where the non-secular DDI H±2 are as follows:
1(cid:88) 1(cid:88)
H(+2) =− D I+I+, H(−2) =− D I−I−. (2)
2 ij i j 2 ij i j
i,j i,j
Here I± =I ±I , I is the projection of the spin angular momentum of spin i on the α axis (α=x,y,z) and D
i ix iy iα ij
is the coupling constant of the DDI between spins i and j. For a linear one-dimensional spin chain D is
ij
γ2(cid:126)(1−3cos2θ)
D = , (3)
ij 2a3|i−j|3
where γ is the gyromagnetic ratio, a is the distance between nearest neighbors, and θ is the angle between the vector
connecting spins i, j and axis z, which is the direction of the strong external magnetic field.
To perform MQ NMR experiments we used quasi-one-dimensional chains of 19F nuclei in calcium fluorapatite
Ca (PO ) F [18]. Experiments were performed on a Brucker Avance III spectrometer with a static magnetic field
5 4 3
B =9.4 T (the corresponding frequency on 19F nuclei is 376.6 MHz). The main block of the MQ NMR experiment
0
(Fig. 1) is repeated many times with the phase increment of the radio-frequency pulses irradiating the spin system
on the preparation period at each repetition. The applied sequence of radio-frequency pulses is built of basic cycles
consisting of eight resonance (for 19F nuclei) π/2 pulses (Fig. 1). At the preparations period of the MQ NMR
experiment, the system is irradiated by several such cycles. At the phase increment of the radio-frequency pulses Φ,
the nonsecular two-spin/two-quantum Hamiltonian (1) has the form (Fig. 1)
H(0) =e−2iΦH(+2)+e2iΦH(−2). (4)
Φ
The phase increments of the radio-frequency pulses serve the purpose of the separation of signals from the MQ NMR
coherences of different orders on the free evolution period [21] (Fig. 1).
SincetheMQNMRcoherencescannotbeobserveddirectlytheyaretransformedintothetransversemagnetization
(single-quantum coherence) in the mixing period. During the mixing period the spin system is irradiated with the
same sequence of the radio-frequency pulses as during the preparation period but the pulse phase is shifted by π/2
(y-pulses are applied instead of x-pulses). Formula (4) shows that the sign of the nonsecular two-spin/two-quantum
Hamiltonian (1) changes due to such a phase shift. The time reversal conditions [22] are necessary for the co-phasing
of contributions of different pairs of spins into the MQ NMR coherences of different orders [1].
The detecting pulse is applied on the detection period for the transmission of the magnetization in the plane
perpendicular to the external magnetic field. The Fourier transform with respect to the phase increment makes it
possible to obtain MQ NMR spectrum consisting of a set of narrow lines corresponding to different coherence orders
[21].
We emphasize that only MQ NMR coherences of the zeroth and plus/minus second orders emerge in the model of
the isolated spin chains in the approximation of the nearest neighbor interactions [12–15]. Our experimental work
[18] demonstrates that more than 97 % of the observed signal corresponds to the MQ NMR coherences of the zeroth
and plus/minus second orders. The theoretical intensities of the MQ NMR coherences of the zeroth (G (τ)) and
0
plus/minus second (G (τ) orders on the preparation period in infinite spin chains are the following [12–15]:
±2
1 1
G (τ)= + J (4Dτ), (5)
0 2 2 0
1 1
G (τ)= − J (4Dτ),
±2 4 4 0
where J is the zero-order Bessel function and D = D = 16.4·103 rad/s is the DDI nearest neighbor coupling
0 i,i+1
constant in the spin chain of 19F nuclei in calcium fluorapatite when the external magnetic field is directed along the
spin chain.
III. A QUANTUM STATE TRANSFER IN ONE-DIMENSIONAL SPIN CHAINS
The ideal quantum state transfer is possible only for homogeneous spin chains consisting of up to 3 spins [23].
However,onecanengineerthecouplingconstants[24]oraddexternalmanipulationofthespinsatthechainends[25]
in order to perform the perfect transfer for chains of arbitrary length. The quantum state transfer in spin chains in
theapproximationofthenearestneighborsisperformedwhenspindynamicsisgovernedbytheflip-flopHamiltonian
H which can be written as
ff
(cid:88)
H = D (I+I− +I−I+ ). (6)
ff i,i+1 i i+1 i i+1
i
Unfortunately, such a Hamiltonian is not the Hamiltonian in MQ NMR experiments. However, the Hamiltonian
(6) is related to the two-spin/two-quantum Hamiltonian (1) by a very simple unitary transformation [10]. Indeed,
performing the transformation U of the Hamiltonian (1)
U =exp(−iπI )·exp(−iπI )..., (7)
2x 4x
We apply selective π pulses which flip the spins in even positions through 180◦ about the x-axis of the rotating
reference frame. As a result, we obtain
(cid:88)
UH(0)U+ = D (I+I− +I−I+ )=H . (8)
i,i+1 i i+1 i i+1 ff
i
Although the transformation (7) cannot be realized experimentally it is very useful for the understanding of peculiar-
ities of the quantum state transfer along spin chains.
The system’s initial state ρ(0), which we consider, has just spin l polarized [26]
1
ρ(0)= ZeβIlz ⊗EN(cid:54)=l, (9)
wheretheparameterβisinverselyproportionaltothetemperature,N isthenumberofspinsinthechain,Z =2Nchβ
2
isthepartitionfunction, l isapositiveoddnumberandE istheidentityoperatorintheHilbertspaceofallspins
N(cid:54)=l
besides spin l. In the MQ NMR experiment the density matrix ρ(t) of the system at time t is
ρ(t)=e−iH(0)tρ(0)eiH(0)t. (10)
The polarization of spin N at time t is
(cid:104)I (cid:105)=Tr{ρ(t)I }=Tr{Uρ(t)I U+}= (11)
Nz Nz Nz
Tr{Uρ(t)U+UINzU+}=Tr{e−iHfftρ(0)eiHfftINz}.
In Eq. (11) we suppose that N is also odd. Thus, we have shown that the quantum state transfer in the MQ NMR
experiment can be described by the flip-flop Hamiltonian of Eq. (6).
The spin dynamics of the considered system is solved exactly in terms of fermionic operators and their Fourier
transform [12, 14]. If the initial polarization was on spin l (see Eq. (9)) than the polarization of spin m at time t is
[27]
(cid:12) (cid:12)2
(cid:104)Imz(cid:105)(t) = 4 (cid:12)(cid:12)(cid:88)exp(−i(cid:15) t)sin(kl)·sin(km)(cid:12)(cid:12) , (12)
(cid:104)I (cid:105)(t) (N +1)2 (cid:12) k (cid:12)
lz (cid:12) (cid:12)
k
where the fermion spectrum (cid:15) is
k
(cid:15) =Dcos(k)+ω , (13)
k 0
where D =D , ω is the Larmor frequency and
i,i+1 0
πn
k = , n=1,2,...N. (14)
N +1
Eq. (12) was obtained [27] in the high temperature approximation [17]. However, it was shown [28] that this
relationship holds at arbitrary temperatures.
Numerical calculations show that almost all polarization (more than 90 %) can be transmitted to the given chain
site for short chains. For long chains the efficiency of transmission gets lower.
The developed approach to the quantum state transfer relies on the approximation of the nearest neighbor inter-
actions. It is known [9] that the DDI of remote spins are not important for the quantum state transfer in short
chains.
IV. RELAXATION OF MQ NMR COHERENCES AS A MODEL OF DECOHERENCE PROCESSES
ThedipolarrelaxationofMQNMRcoherencescanbestudiedontheevolutionperiodoftheMQNMRexperiment.
Relaxation of MQ NMR coherences in one-dimensional systems has been studied earlier [10, 11] using the second
moments of the line shapes of the MQ coherences of the zeroth and second orders. We suggested [29] to perform an
analogous investigation using the ZZ-model, where only the ZZ-part of the DDI is taken into account.
The exact solution [12–15] for MQ NMR dynamics in one-dimensional systems allows us to obtain the density
matrix σ(τ) on the preparation period of the MQ NMR experiment [1] in the approximation of the nearest neighbor
interactions [17, 20] and to write it as follows
σ(τ)=σ (τ)+σ (τ)+σ (τ), (15)
0 2 −2
where σ (τ) (i = 0,2,−2) describes the MQ NMR coherence of order i. If the number of the spins N (cid:29) 1 the
i
contributions σ (τ), σ (τ), σ (τ) are [12, 13]
0 2 −2
1(cid:88)
σ (τ) = cos[2Dτsin(k)](1−a+a ), (16)
0 2 k k
k
1(cid:88)
σ (τ) = − sin[2Dτsin(k)]a a , (17)
2 2 k −k
k
1(cid:88)
σ (τ) = sin[2Dτsin(k)]a+a+ , (18)
−2 2 k −k
k
where k = 2πn, (n=−N/2, −N/2+1, ...N/2−1) and a+, a are the fermion operators [12, 13].
N k k
We study the dipolar relaxation of the MQ NMR coherences [29] on the evolution period of the MQ NMR experi-
ments [1]. This period begins immediately after the preparation period [1] and the density matrix (15) can be used
as the initial state for the relaxation. The relaxation of MQ NMR coherences is caused by the secular (with respect
to the external magnetic field directed along the z-axis) DDI
H =(cid:88)D (3I I −I(cid:126)I(cid:126))=(cid:88)D (2I I −I I −I I ) (19)
dz ij iz jz i j ij iz jz ix jx iy iy
i<j i<j
An investigation of the relaxation process with the Hamiltonian (19) is too complicated. However, the problem is
substantially simplified [29], if we restrict ourselves to the ZZ-part of H only and consider the Hamiltonian
dz
(cid:88) (cid:88)
H =2 D I I = D I I . (20)
ZZ ij iz jz ij iz jz
i<j i(cid:54)=j
The problem with the Hamiltonian H was called [29] the ZZ-model.
ZZ
This approximation is widely used [17] in spin physics (for example, see [30]). The intensity F (τ,t) of the MQ
0
NMR coherence of the zeroth order in the course of the evolution period at the time moment t is
F (τ,t)= Tr{e−iHZZtσ0(τ)eiHZZtσ0(τ)} = Tr{e−iHZZtσ0(τ)eiHZZtσ0(τ)}. (21)
0 Tr(I2) N ·2N−2
z
In Eq. (21) I =(cid:80)N I and F (τ,0)=G (τ) (see Eq. (5)).
z i=1 iz 0 0
Analogously,theintensitiesoftheMQNMRcoherencesoftheplus/minussecondorderontheevolutionperiodare
given by
F (τ,t)= Tr{e−iHZZtσ2(τ)eiHZZtσ−2(τ)}. (22)
±2 Tr(I2)
z
Our calculations demonstrate [29] that the MQ NMR coherence of the zeroth order is not subject to the dipolar
relaxation in the ZZ-model. However, the experimental investigation [29] performed using quasi-one-dimensional
chains of 19F nuclei in calcium fluorapatite Ca (PO ) F shows the dipolar relaxation of the MQ NMR coherence of
5 4 3
the zeroth order due to the flip-flop part of the Hamiltonian (19). The experimental data [29] also demonstrate that
relaxation does not lead to the full disappearance of the MQ NMR of the zeroth order. Relaxation ends with the
stationary intensity of that MQ coherence. The point is that the density matrix of Eq. (16) contains a part which
is proportional to the operator I , commuting with the DDI Hamiltonian (19). As a result, one can obtain [29] the
z
stationary intensity of the MQ NMR coherence of the zeroth order Fst which is
o
J2(2Dτ) 2J2(2Dτ)
Fst = 0 = 0 . (23)
0 G (τ) 1+J (4Dτ)
0 0
The experimental dependence of the stationary intensity of the MQ NMR coherence of the zeroth order during the
preparation period is presented in Fig. 2. The experimental data are in a close agreement with Eq. (21).
Calculations of the relaxation of the MQ NMR coherences of the plus/minus second order with Eq. (21) yield [29]
F (τ,t)= 1 (cid:88) (cid:89) cos[(D +D )t]·[1−(−1)m−m(cid:48)]2J2 (2Dτ). (24)
±2 8N nm nm(cid:48) m−m(cid:48)
m,m(cid:48)n(cid:54)=(m,m(cid:48))
We have shown [29] that the decay of the MQ NMR coherences of the second order conforms reasonably well to a
Gaussian function S(t)=exp{−M2(τ)t2}, where the second moment M (τ) of the line shape of this coherence is
2 2
M2(τ)=− G1(τ)d2F±d2t2(τ,t)(cid:12)(cid:12)(cid:12)(cid:12) (25)
2 t=0
InFig. 3thedependenciesoftheexperimentalandtheoretical(atN =150)dipolarrelaxationtimesonthedurationof
thepreparationperiodareshown. Onecanseethatthetheoreticalpredictionssatisfactorilydescribetheexperimental
data.
FIG. 2: The stationary intensity of the MQ NMR coherence of the zeroth order on the preparation period. The solid line is
the theoretical plot of Eq. (23).
FIG. 3: The experimental times of the dipolar relaxation of the MQ NMR coherence of the second order versus the duration
(cid:113)
of the preparation period τ. The solid line shows the theoretical times t of the dipolar relaxation obtained as t = 2 ,
e e M2(τ)
where the second moment M is determined by Eq. (25).
2
V. CONCLUSIONS
We considered the methods of MQ NMR spectroscopy of one-dimensional spin systems and their application for
solving problems of quantum informatics. The quantum state transfer in linear spin chains can be realized with high
fidelity using MQ NMR dynamics. Decoherence of many-qubit quantum state created on the preparation period of
the MQ NMR experiment [1] can be studied on the evolution period of that experiment. The dipolar relaxation of
theMQNMRcoherencesisconsideredasamodelofthedecoherenceprocess. Thetheoryofthedipolarrelaxationof
the MQ NMR coherences is developed. The performed MQ NMR experiments on the quasi-one-dimensional system
of 19F nuclei in calcium fluorapatite Ca (PO ) F are in a good agreement with the theoretical predictions.
5 4 3
Acknowledgments
The work is supported by the Russian Foundation for Basic Research (Grants 16-03-00056 and 16-33-00867) and
the Program of RAS ”Element base of quantum computers” (Grant No 0089-2015-0220).
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